Before you get started with any of the numerous quests on Eos, you might as well see to the long-ignored outpost of Prodromos, where you’ll naturally find numerous quests you can pick up. Travel back to Eos and when you land at Prodromos head southwest from the Tempest towards the massive water purification rings, near which are two structures. Enter the one closest to the rings and interact with a terminal inside to find the post "Please Help", which details the Darwin-award winning antics of some colonists. This will start the quest "Shock Treatment", which will lead you around Prodromos (see the page for this quest for more in-depth information).
Return to the Tempest and from there make your way south, southeast to reach another shelter, outside of which you’ll find Jacob Mittney. Talk to him and, he should give you the quest "Task: Pathfinder Armor Crafting". While you’re over here, search the nearby Terminal and read the post "Lost Drones" to start the quest "Task: Drone Recovery", which requires you to pretty thoroughly explore Eos.
After you’re done dealing with Jacob Mittney, talk to the "Supply Officer" - Kim Connor - on the platform around the structure he stood near if you wish, then search the building she’s working in front of to find a container. In another building to the east, southeast you’ll find a terminal with a message from Vetra, for what that’s worth.
Leave this building and head south to another, multi-floor structure (or southeast from where you talked to Kim Connor, if you prefer). Enter the bottom floor structure and use a terminal to read the post "Formal request for Darket Tiervian, #28", which starts the quest "Task: Waking up to the Future". The post "Prodromos supplies report", meanwhile, will start the quest "Task: Supply Loss and Recovery", another "hunt the navpoint" fetch quest that won’t take you quite as far out as "Task: Drone Recovery" will. You’ll get to those quests in a bit, but first, consider perusing the wares of the Merchant to the southeast.
Exit the building you’re in and, when you’re outside on the porch, turn northeast head up several flights of stairs and follow the exterior superstructure to reach the upper building. In here you can find Cain Fawkes, who has an awful lot to say, very little of it being worth your time to listen to. On the other hand, a Datapad to the north, northeast contains the details of a duty roster, one of the bits of information you need to save the Prodromos settlers from their own stupidity, if you want to advance the quest "Shock Therapy".
Your next goal is a building to the southwest, which also has two levels. On the ground floor you’ll find two buildings, one to the east, southeast, while the other is to the south, southwest. In the former building you’ll find two people to talk to - Hainly Abrams and Jun-Seo Park - as well as two units of Lab Equipment to scan. Depending on your progress through the game, Hainly Abrams might mention some worrisome tremors, ultimately starting the quest "Making an Impression". In the other building you’ll find Majordomo Bell-Scott who talks a lot, and should probably switch to decaffeinated coffee. If you complete the quest "Task: Missing Colonists" he should give you a related quest "Doesn’t Add Up" - something to keep in mind.
Leave the Majordomo’s office, turn south, southwest and jump up to the roof of the building you were just in, then head upstairs into the nearby building. Behold, the med-bay! In here you’ll find Chief Lito, if you woke her from cryo, although she has little of interest to say. Continue into the next room where you’ll find a terminal to mess with, Dr. Ramirez to talk to, and most importantly scan the Medical Equipment near him to learn more about the colonists who are attempting to commit suicide-by-Remnant.
Before you depart from Prodromos on your first of several Prodromos-based quests, leave the building and head west to find August Bradley, the mayor of this little outpost. Get through his initial dialogue, then chat with him again to get a traditional wheel of dialogue options populated with various small-talk options. Some of it’s interesting stuff, as far as flavor and backstory goes, but nothing he says is essential, quest-wise.
Now it’s time to get on to questing… might as well start with some of old oldest quests first, work your way from quests that start out on the Nexus to quest that start in Prodromos, ultimately ending with new quests that trigger west of Prodromos. More or less, anyways